2016 Special courses on advanced topics in Mathematics B

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Academic unit or major
Mathematics
Instructor(s)
Ma Shohei 
Course component(s)
Lecture
Day/Period(Room No.)
Intensive (H201)  
Group
-
Course number
ZUA.E332
Credits
2
Academic year
2016
Offered quarter
3Q
Syllabus updated
2017/1/11
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

The fundamental concepts are:
Riemann-Roch theorem, Serre duality, canonical model, Hodge decomposition, Torelli theorem
Solve the exercises presented in the lecture.

Student learning outcomes

We will study the basic theory of complex algebraic curves. There are two main topics: (1) the Hodge decomposition of the cohomology, which leads to the Jacobian, the Abel-Jacobi map and the Torelli theorem; and (2) the canonical model which is studied using the cohomology of line bundles. These two topics are closely related by the Gauss map of the Abel-Jacobi image.

Keywords

Algebraic curves, Riemann surfaces

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 Algebraic curves Details will be provided during each class session
Class 2 Line bundles Details will be provided during each class session
Class 3 Cohomology Details will be provided during each class session
Class 4 Riemann-Roch formula Details will be provided during each class session
Class 5 Serre duality Details will be provided during each class session
Class 6 Canonical model Details will be provided during each class session
Class 7 Clifford's theorem Details will be provided during each class session
Class 8 Green conjecture Details will be provided during each class session
Class 9 Jacobi variety Details will be provided during each class session
Class 10 Abel-Jacobi map Details will be provided during each class session
Class 11 Torelli theorem Details will be provided during each class session
Class 12 Theta divisor Details will be provided during each class session
Class 13 Moduli space Details will be provided during each class session
Class 14 Canonical divisor of Mg Details will be provided during each class session
Class 15 Harris-Mumford theorem Details will be provided during each class session

Textbook(s)

None specified.

Reference books, course materials, etc.

E.Arbarello, M.Cornalba, P.Griffiths, J.,Harris, `Geometry of Algebraic Curves I' Springer.
R.Narashimhan, `Compact Riemann surfaces'
J.Harris, I.Morrison, `Moduli of Curves' Springer

Assessment criteria and methods

By report. Details will be provided during each class session.

Related courses

  • ZUA.C301 : Complex Analysis I
  • MTH.C302 : Complex Analysis II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None in particular

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